FMSP Lectures

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2015/10/22

17:00-17:50   Room #002 (Graduate School of Math. Sci. Bldg.)
Hans-Otto Walther (University of Giessen)
Shilnikov chaos due to state-dependent delay, by means of the fixed point index (ENGLISH)
[ Abstract ]
What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional d(¥phi), with d(¥phi)=1 on a neighborhood of ¥phi=0, such that the equation x'(t)=-a x(t-d(x_t)) has a solution which is homoclinic to 0, with shift dynamics in its vicinity, whereas the linear equation x'(t)=-a x(t-1) with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.
The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf